A modular k-coloring, k³2, of a graph G without isolated vertices is a coloring of the vertices of G with the elements in Zk having the property that for every two adjacent vertices of G, the sums of the colors of the neighbors are different in Zk. The minimum k for which G has a modular k-coloring is the modular Chromatic Number of G. Except for some special cases modular Chromatic Number of Cm Pn is determined.

Let $G$ be a simple connected graph having finite Number of vertices (nodes). Let a coloring game is played on the nodes of $G$ by two players, Alice and Bob alternately assign colors to the nodes such that the adjacent nodes receive different colors with Alice taking first turn. Bob wins the game if he is succeeded to assign k distinct colors in the neighborhood of some vertex, where k is the available Number of colors. Otherwise, Alice wins. The game Chromatic Number of G is the minimum Number of colors that are needed for Alice to win this coloring game and is denoted by $\chi_{g}(G)$. In this paper, the game Chromatic Number $\chi_{g}(G)$ for some interconnecting networks such as infinite honeycomb network, elementary wall of infinite height and infinite octagonal network is determined. Also, the bounds for the game Chromatic Number $\chi_{g}(G)$ of infinite oxide network are explored.

A {\it local antimagic labeling} of a connected graph GG with at least three vertices, is a bijection f: E(G)→ {1, 2, … , |E(G)|}f: E(G)→ {1, 2, … , |E(G)|} such that for any two adjacent vertices uu and vv of GG, the condition ω f(u)≠ ω f(v)ω f(u)≠ ω f(v) holds; where ω f(u)=∑ x∈ N(u)f(xu)ω f(u)=∑ x∈ N(u)f(xu). Assigning ω f(u)ω f(u) to uu for each vertex uu in V(G)V(G), induces naturally a proper vertex coloring of GG; and |f||f| denotes the Number of colors appearing in this proper vertex coloring. The {\it local antimagic Chromatic Number} of GG, denoted by χ la(G)χ la(G), is defined as the minimum of |f||f|, where ff ranges over all local antimagic labelings of GG. In this paper, we explicitly construct an infinite class of connected graphs GG such that χ la(G)χ la(G) can be arbitrarily large while χ la(G∨ K2¯ )=3χ la(G∨ K2¯ )=3, where G∨ K2¯ G∨ K2¯ is the join graph of GG and the complement graph of K2K2. The aforementioned fact leads us to an infinite class of counterexamples to a result of [Local antimagic vertex coloring of a graph, Graphs and Combinatorics 33} (2017), 275-285].

‎For a graph G=(V,E) and a vertex subset $D\subseteq V$‎, ‎a vertex $v\in V$ is called a dominator of D if v is adjacent to every vertex in D‎, ‎and an anti-dominator of D if v is not adjacent to any vertex in D. ‎Given a coloring $C=\{V_{1},V_{2},\ldots,V_{k}\}$ of $G$‎, ‎a color {class $V_{i}$} {is a dominating color class (resp. an anti dominating color class) for a vertex ‎v‎‎ if ‎‎v‎‎ dominates all vertices in ‎$‎V_i‎$‎ (resp. ‎‎v‎‎ dominates no vertex in ‎$‎V_i‎$‎)}‎. ‎A coloring C is a global dominator coloring if each vertex in $G$ has both a dominating and an anti-dominating color class‎. ‎The global dominator Chromatic Number‎, ‎denoted by $\chi_{gd}(G)$‎, ‎is the minimum Number of colors required for a global dominator coloring of $G$‎. ‎In this paper‎, ‎we investigate the global dominator Chromatic Number for various classes of graphs‎.

A graph $G$ of order $n$ is called $k-$step Hamiltonian for $k\geq 1$ if we can label the vertices of $G$ as $v_1,v_2,\ldots,v_n$ such that $d(v_n,v_1)=d(v_i,v_{i+1})=k$ for $i=1,2,\ldots,n-1$. The (vertex) Chromatic Number of a graph $G$ is the minimum Number of colors needed to color the vertices of $G$ so that no pair of adjacent vertices receive the same color. The clique Number of $G$ is the maximum cardinality of a set of pairwise adjacent vertices in $G$. In this paper, we study the Chromatic Number and the clique Number in $k-$step Hamiltonian graphs for $k\geq 2$. We present upper bounds for the Chromatic Number in $k-$step Hamiltonian graphs and give characterizations of graphs achieving the equality of the bounds. We also present an upper bound for the clique Number in $k-$step Hamiltonian graphs and characterize graphs achieving equality of the bound.

Let G be a simple graph. The dominated coloring of G is a proper coloring of G such that each color class is dominated by at least one vertex. The minimum Number of colors needed for a dominated coloring of G is called the dominated Chromatic Number of G, denoted by xdom(G). Stability (bondage Number) of dominated Chromatic Number of G is the minimum Number of vertices (edges) of G whose removal changes the dominated Chromatic Number of G. In this paper, we study the dominated Chromatic Number, dominated stability and dominated bondage Number of certain graphs.

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A local antimagic edge labeling of a graph $G=(V,E)$ is a bijection $f:E\rightarrow\{1,2,\dots,|E|\}$ such that the induced vertex labeling $f^+:V\rightarrow \mathbb{Z}$ given by $f^+(u)=\sum f(e),$ where the summation runs over all edges $e$ incident to $u,$ has the property that any two adjacent vertices have distinct labels. A graph $G$ is said to be locally antimagic if it admits a local antimagic edge labeling. The local antimagic Chromatic Number $\chi_{la}(G)$ is the minimum Number of distinct induced vertex labels over all local antimagic  labelings of $G.$ In this paper we obtain sufficient conditions under which $\chi_{la}(G\vee H),$ where $H$ is either a cycle or the empty graph $O_n=\overline{K_n},$ satisfies a sharp upper bound. Using this we determine the value of $\chi_{la}(G\vee H)$ for many wheel related graphs $G.$

For any simple graph $G$, the signless Laplacian matrix of $G$ is defined as $D(G)+A(G)$, where $D(G)$ and $A(G)$ are the diagonal matrix of vertex degrees and the adjacency matrix of $G$, respectively. %Let $\chi(G)$ be the Chromatic Number of $G$ Let $q(G)$ be the signless Laplacian spectral radius of $G$ (the largest eigenvalue of the signless Laplacian matrix of $G$). In this paper we find some relations between the Chromatic Number and the signless Laplacian spectral radius of graphs. In particular, we characterize all graphs $G$ of order $n$ with odd Chromatic Number $\chi$ such that $q(G)=2n\Big(1-\frac{1}{\chi}\Big)$. Finally we show that if $G$ is a graph of order $n$ and with Chromatic Number $\chi$, then under certain conditions, $q(G)<2n\Big(1-\frac{1}{\chi}\Big)-\frac{2}{n}$. This result improves some previous similar results.

An edge labeling of a connected graph G = (V, E) is said to be local antimagic if it is a bijection f: E →{1, . . ., |E|} such that for any pair of adjacent vertices x and y, f+(x)≠ f+(y), where the induced vertex label f+(x)= ∑ f(e), with e ranging over all the edges incident to x. The local antimagic Chromatic Number of G, denoted by Xla(G), is the minimum Number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, the sharp lower bound of the local antimagic Chromatic Number of a graph with cut-vertices given by pendants is obtained. The exact value of the local antimagic Chromatic Number of many families of graphs with cut-vertices (possibly given by pendant edges) are also determined. Consequently, we partially answered Problem 3. 1 in [Local antimagic vertex coloring of a graph, Graphs and Combin., 33, (2017), 275--285].